Hierarchical)&)Spectral)clustering)...
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Hierarchical & Spectral clustering Lecture 13 David Sontag New York University Slides adapted from Luke Zettlemoyer, Vibhav Gogate, Carlos Guestrin, Andrew Moore, Dan Klein
Transcript of Hierarchical)&)Spectral)clustering)...
Hierarchical)&)Spectral)clustering)Lecture)13)
David&Sontag&New&York&University&
Slides adapted from Luke Zettlemoyer, Vibhav Gogate, Carlos Guestrin, Andrew Moore, Dan Klein
Agglomerative Clustering • Agglomerative clustering:
– First merge very similar instances – Incrementally build larger clusters out
of smaller clusters
• Algorithm: – Maintain a set of clusters – Initially, each instance in its own
cluster – Repeat:
• Pick the two closest clusters • Merge them into a new cluster • Stop when there’s only one cluster left
• Produces not one clustering, but a family of clusterings represented by a dendrogram
Agglomerative Clustering • How should we define �closest� for clusters
with multiple elements?
Agglomerative Clustering • How should we define �closest� for clusters
with multiple elements?
• Many options: – Closest pair
(single-link clustering) – Farthest pair
(complete-link clustering) – Average of all pairs
• Different choices create different clustering behaviors
Agglomerative Clustering • How should we define �closest� for clusters
with multiple elements?
Farthest pair (complete-link clustering)
Closest pair (single-link clustering) Single Link Example
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Complete Link Example
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[Pictures from Thorsten Joachims]
Clustering&Behavior&Average
Mouse tumor data from [Hastie et al.]
Farthest Nearest
Agglomera<ve&Clustering&
When&can&this&be&expected&to&work?&
Closest pair (single-link clustering) Single Link Example
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Strong separation property: All points are more similar to points in their own cluster than to any points in any other cluster
Then, the true clustering corresponds to some pruning of the tree obtained by single-link clustering!
Slightly weaker (stability) conditions are solved by average-link clustering
(Balcan et al., 2008)
Spectral)Clustering)
Slides adapted from James Hays, Alan Fern, and Tommi Jaakkola
Spectral)clustering)
[Shi & Malik ‘00; Ng, Jordan, Weiss NIPS ‘01]
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K-means Spectral clustering
Spectral)clustering)
[Figures from Ng, Jordan, Weiss NIPS ‘01]
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Spectral)clustering)
))Group)points)based)on)links)in)a)graph)
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[Slide from James Hays]
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��
(y takes discrete values)
• Relax�the�optimization�problem�into�the�continuous�domain�by�solving�generalized�eigenvalue�system:
���" �� � ' � � subject�to����� ( �
• Which�gives: � ' � � ( ���• Note�that� � ' � � ( �,�so�the�first�eigenvector�is��� ( �
with�eigenvalue��.• The�second�smallest�eigenvector�is�the�real�valued�solution�to�
this�problem!!
Solving�NCut
2�way�Normalized�Cuts
1. Compute�the�affinity�matrix�W,�compute�the�degree�matrix�(D),�D�is�diagonal�and�
!��2. Solve� ,�where� is�
called�the�Laplacian matrix3. Use�the�eigenvector�with�the�second�smallest�
eigen�value�to�bipartition�the�graph�into�two�parts.
Creating�Bi�partition�Using�2ndEigenvector
• Sometimes�there�is�not�a�clear�threshold�to�split�based�on�the�second�vector�since�it��takes�continuous�values
• How�to�choose�the�splitting�point?�a) Pick�a�constant�value�(0,�or�0.5).b) Pick�the�median�value�as�splitting�point.c) Look�for�the�splitting�point�that�has�the�minimum�Ncut
value:1. Choose�n possible�splitting�points.2. Compute�Ncut value.3. Pick�minimum.
Spectral clustering: example
−3 −2 −1 0 1 2 3 4 5−2
−1
0
1
2
3
4
5
6
−4 −2 0 2 4 6−2
−1
0
1
2
3
4
5
6
Tommi Jaakkola, MIT CSAIL 18
Spectral clustering: example cont’d
0 5 10 15 20 25 30 35 40−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Components of the eigenvector corresponding to the secondlargest eigenvalue
Tommi Jaakkola, MIT CSAIL 19
K�way�Partition?
• Recursive�bi�partitioning�(Hagen�et�al.,̂ 91)– Recursively�apply�bi�partitioning�algorithm�in�a�hierarchical�divisive�manner.
– Disadvantages:�Inefficient,�unstable• Cluster�multiple�eigenvectors– Build�a�reduced�space�from�multiple�eigenvectors.– Commonly�used�in�recent�papers– A�preferable�approach`�its�like�doing�dimension�reduction�then�k�means
![A Tutorial on Spectral Clustering - Max Planck Society1].… · A Tutorial on Spectral Clustering Ulrike von Luxburg Abstract. In recent years, spectral clustering has become one](https://static.fdocuments.net/doc/165x107/5ba91ad009d3f2810a8bc19c/a-tutorial-on-spectral-clustering-max-planck-1-a-tutorial-on-spectral-clustering.jpg)
![Spectral Curvature Clustering for Hybrid Linear Modeling · Our algorithm, Spectral Curvature Clustering (SCC), combines Govindu’s frame-work [19] and Ng et al.’s spectral clustering](https://static.fdocuments.net/doc/165x107/6017b0c3eac3e56f30301ddd/spectral-curvature-clustering-for-hybrid-linear-modeling-our-algorithm-spectral.jpg)
